Integral of \( \displaystyle \left(3 x - 1\right) \sin{\left(3 x - 1 \right)} \)
Problem 4.1478 · medium Video
Find \( \displaystyle \int \left(3 x - 1\right) \sin{\left(3 x - 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(3 x - 1\right) \sin{\left(3 x - 1 \right)}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = \frac{\left(1 - 3 x\right) \cos{\left(3 x - 1 \right)}}{3} - \int \left(- \cos{\left(3 x - 1 \right)}\right)\, dx \]parts simplifyApply integration by parts with u = 3*x - 1 and dv = sin(3*x - 1)dx. Simplify the integrand in the remaining integral.✓ Proved
- \[ = \frac{\left(1 - 3 x\right) \cos{\left(3 x - 1 \right)}}{3} + \int \cos{\left(3 x - 1 \right)}\, dx \]algebraDistribute the negative sign.✓ Proved
- \[ = \frac{\left(1 - 3 x\right) \cos{\left(3 x - 1 \right)}}{3} + \frac{\sin{\left(3 x - 1 \right)}}{3} \]antiderivativeEvaluate the final integral.✓ Proved
Answer \( - x \cos{\left(3 x - 1 \right)} + \frac{\sin{\left(3 x - 1 \right)}}{3} + \frac{\cos{\left(3 x - 1 \right)}}{3} + C \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: fail (error) — The final antiderivative given in the stated answer contains an extra +cos(3*x‑1)/3 term. The correct result, obtained by the steps shown, is -(3*x‑1)*cos(3*x‑1)/3 + sin(3*x‑1)/3, so the answer is mathematically incorrect.qwen3.6:27b-mlx: pass — The solution correctly applies integration by parts, simplifies the resulting expression, and evaluates the remaining integral. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-09 — The solution correctly applies integration by parts, simplifies the resulting expression, and evaluates the remaining integral. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-11-09 — The final antiderivative given in the stated answer contains an extra +cos(3*x‑1)/3 term. The correct result, obtained by the steps shown, is -(3*x‑1)*cos(3*x‑1)/3 + sin(3*x‑1)/3, so the answer is mathematically incorrect.qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly applies integration by parts and subsequent simplification steps. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-08qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly applies integration by parts, simplifies the resulting expression, and evaluates the remaining integral. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-08
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-11-09 with SymPy 1.14.0.